3. Reference problem: Multi-fiber beams#

3.1. Geometry and meshes#

We consider a beam with a length \(2m\) and a square cross section with a side \(0.2m\).

_images/10000000000001C70000007540AD4BFBD08C4EF1.png

For multifibre beam models, the section of the beam includes 16 fibers arranged as in the following figure:

_images/1000000000000212000001ECAC6083E05EA76341.png

This same beam is also represented by 3D geometry.

_images/100000000000025E0000007C29CFFBB9A982D053.png

3.2. Material properties#

The materials are linear elastic:

Concrete: \(E\mathrm{=}30\mathit{GPa}\) steel: \(E\mathrm{=}200\mathit{GPa}\)

\(\nu \mathrm{=}0.2\) \(\nu \mathrm{=}0.3\)

\(\alpha \mathrm{=}{10}^{\mathrm{-}5}{K}^{\mathrm{-}1}\) \(\alpha \mathrm{=}2.{10}^{\mathrm{-}5}{K}^{\mathrm{-}1}\)

In the first calculation (mono-material), the beam is entirely made of concrete.

In the second and third calculations (bi-material), the fibers under the \(Y\) axis are concrete while the fibers above are steel.

3.3. Boundary conditions and loading#

3.3.1. Boundary conditions#

Beams calculations 1 and 2:

Node P1 is stuck in DX, DY, DZ, DRX, and DRZ.

Node P2 is locked in DY and DZ.

Beams calculation 3:

The nodes P 1 and P 2 are embedded.

3D:

The nodes on segment ENC_0 are locked in DX and DZ.

The nodes on segment ENC_1 are locked in DZ.

The nodes P0 and P1 are locked in DY.

3.3.2. Loading#

Mono-material calculation:

A thermal loading dependent on \(Z\) and on the time \(t\) is imposed. Temperature is defined as: \(T\mathrm{=}(\mathrm{-}200Z+20)t\). The reference temperature is \(0\).

Bi-material calculations:

A thermal loading dependent on \(Z\) and on the time \(t\) is imposed. Temperature is defined as follows:

  • \(T\mathrm{=}20t\) if \(Z>0\)

  • \(T\mathrm{=}40t\) if \(Z<0\)

The reference temperature is \(0\).